To stabilize PDE models, control laws require space-dependent functional gains mapped by nonlinear operators from the PDE functional coefficients. When a PDE is nonlinear and its "pseudo-coefficient" functions are state-dependent, a gain-scheduling (GS) nonlinear design is the simplest approach to the design of nonlinear feedback. The GS version of PDE backstepping employs gains obtained by solving a PDE at each value of the state. Performing such PDE computations in real time may be prohibitive. The recently introduced neural operators (NO) can be trained to produce the gain functions, rapidly in real time, for each state value, without requiring a PDE solution. In this paper we introduce NOs for GS-PDE backstepping. GS controllers act on the premise that the state change is slow and, as a result, guarantee only local stability, even for ODEs. We establish local stabilization of hyperbolic PDEs with nonlinear recirculation using both a "full-kernel" approach and the "gain-only" approach to gain operator approximation. Numerical simulations illustrate stabilization and demonstrate speedup by three orders of magnitude over traditional PDE gain-scheduling. Code (Github) for the numerical implementation is published to enable exploration.

翻译：为稳定偏微分方程模型，控制律需要由非线性算子从偏微分方程函数系数映射出的空间相关函数增益。当偏微分方程非线性且其“伪系数”函数依赖于状态时，增益调度非线性设计是实现非线性反馈的最简单方法。偏微分方程反步法的增益调度版本通过在每个状态值下求解偏微分方程来获取增益。实时执行此类偏微分方程计算可能代价高昂。近期引入的神经算子可被训练用于在实时状态下快速生成各状态值的增益函数，而无需求解偏微分方程。本文针对增益调度-偏微分方程反步法引入神经算子。增益调度控制器基于状态变化缓慢的前提运作，因此即使对于常微分方程，也只能保证局部稳定性。我们通过“全核”方法和“仅增益”方法进行增益算子逼近，建立了具有非线性再循环的双曲型偏微分方程的局部镇定。数值仿真展示了镇定效果，并表明相较于传统偏微分方程增益调度，加速效果达到三个数量级。数值实现的代码已在GitHub上发布以供探索。